LEONARDTOWN, Md.—To the outside world, it came across as mood swings and anger. But Regina Cullison would later be told by psychiatrists she struggled with depression and anxiety—and that she needed help. And that is where her trouble began and ended.

According to her mother, there were few psychiatrists in the county who took private insurance. When Ms. Cullison lost her job as a dentist's assistant, and with it her insurance, she...